Efficient prediction for linear and nonlinear autoregressive models. (English) Zbl 1106.62103

Müller, Ursula U.; Schick, Anton; Wefelmeyer, Wolfgang

Ann. Stat. 34, No. 5, 2496-2533 (2006).

Summary: Conditional expectations given past observations in stationary time series are usually estimated directly by kernel estimators, or by plugging in kernel estimators for transition densities. We show that, for linear and nonlinear autoregressive models driven by independent innovations, appropriate smoothed and weighted von Mises statistics of residuals estimate conditional expectations at better parametric rates and are asymptotically efficient. The proof is based on a uniform stochastic expansion for smoothed and weighted von Mises processes of residuals. We consider, in particular, estimation of conditional distribution functions and of conditional quantile functions.

MSC:

62M20	Inference from stochastic processes and prediction
62G20	Asymptotic properties of nonparametric inference
62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G08	Nonparametric regression and quantile regression
60F17	Functional limit theorems; invariance principles

Keywords:

empirical likelihood; Owen estimator; weighted density estimator; kernel smoothed empirical process; functional central limit theorem; Donsker class; uniformly integrable entropy; uniformly integrable bracketing entropy, pseudo-observation; plug-in-estimator; AR model; EXPAR model; SETAR model

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